Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,21)(12,15)(13,23)(14,19)(16,20)(17,18)(22,24);(1,3,5,4)(2,6,7,8)(9,11,15,22,16,23,14,18)(10,17,19,13,20,24,12,21);(1,4,5,3)(2,6,7,8)(9,12,14,20,16,19,15,10)(11,17,18,21,23,24,22,13);(2,7)(6,8)(9,13,15,24,16,21,14,17)(10,18,19,11,20,22,12,23)Polycyclic presentation:
2^8|b^2=eh,c^2=e,d^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26854, #6443962, #6443964, … |
| maximal normal subgroup of | C2×#26854, #6443962, #6443964, … |
| maximal quotient of | #311137, #311140, #311142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19842);